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Integers that are sums of two rational sixth powers
Alexis Newton, Jeremy Rouse
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We prove that $164634913$ is the smallest positive integer that is a sum of two rational sixth powers but not a sum of two integer sixth powers. If Cₖ is the curve x⁶ + y⁶ = k, we use the existence of morphisms from Cₖ to elliptic curves, together with the Mordell-Weil sieve, to rule out the existence of rational points on Cₖ for various k.
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